Note that this can alternatively be proved by analyzing the Geršgorin discs; see for example [1]}. Had we not taken the expectation in (REF ) and symmetrized the weighted adjacency matrix in (REF ), we would have needed a stronger bound here [1]}.

Let \(u_1\)  be an eigenvector corresponding to the largest eigenvalue \(\lambda _1\)  of \(w\)  with \(\left\Vert u_1\right\Vert =1\) .
By the Rayleigh quotient theorem [1]},
\(v^T w v \le u_1^T w u_1=\lambda _1\) 
